The null space of the map to diagonal operator has a special structure, it is a linear subspace which is closed under taking adjoints. This observation led us to consider the following notions for the first time in the context of LOCC state distinguishability. First we recall the basic structure theory for finite-dimensional \(C^*\) -algebras, for instance as exhibited in [1]}, [2]}. Every such algebra is unitarily equivalent to an orthogonal direct sum of ampliated full matrix algebras of the form \(\bigoplus _{k}(M_{m_k}(\mathbb {C})\otimes I_{n_k})\)  for some positive integers \(m_k, n_k \ge 1\) . Further, the algebra is unital if it contains the identity operator.
